IODS course project
Riku Turkki
About the project
In this project I’ll learn the basics of MarkDown and git/GitHub. Here’s a link to my IODS-project GitHub page
Regression and model validation
Haven’t done this excercise :(
Logistic regression
Haven’t done this excercise ):
Clustering and classification
The topics of this chapter - clustering and classification - are handy and visual tools of exploring statistical data. Clustering means that some points (or observations) of the data are in some sense closer to each other than some other points. In other words, the data points do not comprise a homogeneous sample, but instead, it is somehow clustered.
In general, the clustering methods try to find these clusters (or groups) from the data. One of the most typical clustering methods is called k-means clustering. Also hierarchical clustering methods quite popular, giving tree-like dendrograms as their main output.
As such, clusters are easy to find, but what might be the “right” number of clusters? It is not always clear. And how to give these clusters names and interpretations?
Based on a successful clustering, we may try to classify new observations to these clusters and hence validate the results of clustering. Another way is to use various forms of discriminant analysis, which operates with the (now) known clusters, asking: “what makes the difference(s) between these groups (clusters)?”
In the connection of these methods, we also discuss the topic of distance (or dissimilarity or similarity) measures. There are lots of other measures than just the ordinary Euclidean distance, although it is one of the most important ones. Several discrete and even binary measures exist and are widely used for different purposes in various disciplines.
Point 2
## crim zn indus chas nox rm age dis rad tax ptratio black
## 1 0.00632 18 2.31 0 0.538 6.575 65.2 4.0900 1 296 15.3 396.90
## 2 0.02731 0 7.07 0 0.469 6.421 78.9 4.9671 2 242 17.8 396.90
## 3 0.02729 0 7.07 0 0.469 7.185 61.1 4.9671 2 242 17.8 392.83
## 4 0.03237 0 2.18 0 0.458 6.998 45.8 6.0622 3 222 18.7 394.63
## 5 0.06905 0 2.18 0 0.458 7.147 54.2 6.0622 3 222 18.7 396.90
## 6 0.02985 0 2.18 0 0.458 6.430 58.7 6.0622 3 222 18.7 394.12
## lstat medv
## 1 4.98 24.0
## 2 9.14 21.6
## 3 4.03 34.7
## 4 2.94 33.4
## 5 5.33 36.2
## 6 5.21 28.7## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...## [1] 506 14Point 3
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00Comments on point 3
Based on the scatter plot it seems that variables rm and lstat have a moderate anticorrelation (r = -0.614) whereas rm and medv seems to have positive correlation (r = 0.695). In addition, variable indus seems to have rather strong correlation with tax and nox (r = 0.721 and r = 0.764 respectively). dis weighted mean of distances to five Boston employment centres.
nox = nitrogen oxides concentration (parts per 10 million).
rm = average number of rooms per dwelling.
tax = full-value property-tax rate per $10,000.
lstat = lower status of the population (percent).
medv = median value of owner-occupied homes in $1000s.
Point 4
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865Comments on point 4
As can be seen from the plot above scaling the data frame obviously preseves the shape and distribution, however, the mean values are now set to zero and standard deviation to 1.
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## intersect, setdiff, setequal, unionPoint 5
Point 6
## predicted
## correct low med_low med_high high
## low 18 4 1 0
## med_low 5 12 14 0
## med_high 1 5 22 2
## high 0 0 1 17## Loading required package: lattice## Confusion Matrix and Statistics
##
## predicted
## correct low med_low med_high high
## low 18 4 1 0
## med_low 5 12 14 0
## med_high 1 5 22 2
## high 0 0 1 17
##
## Overall Statistics
##
## Accuracy : 0.6765
## 95% CI : (0.5766, 0.7658)
## No Information Rate : 0.3725
## P-Value [Acc > NIR] : 4.815e-10
##
## Kappa : 0.5639
## Mcnemar's Test P-Value : NA
##
## Statistics by Class:
##
## Class: low Class: med_low Class: med_high Class: high
## Sensitivity 0.7500 0.5714 0.5789 0.8947
## Specificity 0.9359 0.7654 0.8750 0.9880
## Pos Pred Value 0.7826 0.3871 0.7333 0.9444
## Neg Pred Value 0.9241 0.8732 0.7778 0.9762
## Prevalence 0.2353 0.2059 0.3725 0.1863
## Detection Rate 0.1765 0.1176 0.2157 0.1667
## Detection Prevalence 0.2255 0.3039 0.2941 0.1765
## Balanced Accuracy 0.8429 0.6684 0.7270 0.9413Comments on point 6
On overall, the classification with LDA seems to be working quite nicely. Classification accuracy is 0.765 (95% CI 0.6704, 0.8431) and Kappa value of 0.68 indicates substantial agreement.
Point 7
Comments on point 7
Based on the plot above k=2 is the optimal number of clusters.
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## Warning in cor(x, y, method = method, use = use): the standard deviation is
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## Warning in cor(x, y, method = method, use = use): the standard deviation is
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## Warning in cor(x, y, method = method, use = use): the standard deviation is
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## Warning in cor(x, y, method = method, use = use): the standard deviation is
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## Warning in cor(x, y, method = method, use = use): the standard deviation is
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## Warning in cor(x, y, method = method, use = use): the standard deviation is
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Comments on point 7 contd.
Most obvious discriminating variables, bases on the plot above, are at least zn, nox, and tax. Blue group indicates cases where crime rates are higher and in red group the crime rates seems to be usually lower.
Bonus
Comments on Bonus
Strongest predictor variables are nox, tax, zn, and medv
Super-Bonus
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## layout## [1] 404 13## [1] 13 3Comments on super-bonus
Clustering (k=4) is not a perfect match with the real crime categories but is resembels it remarcably well.
Comments on point 2
The dataset contains 506 obersvations and 14 variables describing crime statistic in Boston area. Description of dataset variables ca be found from here